An algebraic approach to laying a ghost to rest

Abstract

In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck \cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {\em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim \cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of $\mathcal{P}\mathcal{T}$ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Dirac's theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order \cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.

8 pages

References in corpus (3)

• Solution to the ghost problem in fourth order derivative theories
• Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator
• Lagrangians Galore

Cited by in corpus (9)

• Stable Self-Interacting Pais-Uhlenbeck Oscillator
• Pais-Uhlenbeck Oscillator and Negative Energies
• Pais-Uhlenbeck Oscillator with a Benign Friction Force
• On Negative Energies, Strings, Branes, and Braneworlds: A Review of Novel Approaches
• Noether symmetries and the quantization of a Lienard-type nonlinear oscillator
• Natural Higher-Derivatives Generalization for the Klein-Gordon Equation
• Quantization of quadratic Liénard-type equations by preserving Noether symmetries
• Markov constant and quantum instabilities
• Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model